Entire solutions blowing up at infinity for semilinear elliptic systems

We consider the system Δu=p(x)g(v), Δv=q(x)f(u) in $\text{R}{}^{\mathrm{N}}$, where f,g are positive and non-decreasing functions on (0,∞) satisfying the Keller–Osserman condition and we establish the existence of positive solutions that blow-up at infinity.     

Régularité dans une équation de Schrödinger avec potentiel singulier à distance finie et à l'infini

In this Note we study the Schrödinger equation i?_tu+Δu+V₀u+V₁u=0 on $\text{R}{}^3\text{×(0,T)}$ with initial condition $\text{u}{}_0\text{∈{v∈H}{}^2\text{(}\text{R}{}^3\text{)}$, where V₀ is a coulombian potential, singular at finite distance and V₁ is an electric potential, possibly unbounded. Both of them may depend on space and time variables. We prove that this problem is well-posed and that the regularity of the initial data is conserved for the solution. The detailed proof will be given elsewhere (Baudouin et al., in press). To cite this article: L. Baudouin et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A comparison result related to Gauss measureUn résultat de comparaison relatif à la mesure de Gauss

In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type

where $\text{Ω}$ is an open set of $\text{R}{}^{\mathrm{n}}$ (n?2), ?(x)=(2π)^?n/2exp(?|x|²/2), a_ij(x) are measurable functions such that a_ij(x)ξ_iξ_j??(x)|ξ|² a.e. $\text{x∈Ω}$, $\text{ξ∈}\text{R}{}^{\mathrm{n}}$ and f(x) is a measurable function taken in order to guarantee the existence of a solution $\text{u∈}\text{H}{}^1{}_0\text{(?,Ω)}$ of (1.1). We use the notion of rearrangement related to Gauss measure to compare u(x) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456.     

Doubly nonlinear parabolic equations with singular lower order term

In this paper we are concerned with positive solutions of the doubly nonlinear parabolic equation u_t=div(u^m−1|∇u|^p−2∇u)+Vu^m+p−2 in a cylinder $\text{Ω×(0,T)}$, with initial condition u(·,0)=u₀(·)⩾0 and vanishing on the parabolic boundary $\text{∂Ω×(0,T)}$. Here $\text{Ω⊂}\text{R}{}^{\mathrm{N}}$ (resp. $\text{H}{}^{\mathrm{n}}$) is a bounded domain with smooth boundary, $\text{V∈L}{}_{\mathrm{<mtext>loc</mtext>}}{}^1\text{(Ω)}$, $\text{m∈}\text{R}$, 1<p<N and m+p−2>0. The critical exponents $\text{q}{}^1$ are found and the nonexistence results are proved for $\text{q}{}^1\text{⩽m+p<3}$.     

Nonstandard points on algebraic curves

Let Γ be an algebraic curve which is given by an equation f(x, y) = 0, f(x, y) ∈ k[x, y] where k is an algebraic number field and f(x, y) is irreducible. Suppose that there exists an ${}^1\text{Γ}$ a nonstandard point $\text{(ξ, η) ∈}{}^1\text{k ×}{}^1\text{k}$. Then k(ξ, η) is (isomorphic to) the algebraic function field of Γ and, at the same time, is a subfield of ${}^1\text{k}$. Correlating the divisors of the function field k(ξ, η) and of the number field ${}^1\text{k}$, we develop an analogue of the Artin-Whaples theory of the product formula. This leads to one of Siegel's basic inequalities for rational points on algebraic curves.     

Positive solutions and large time behaviors of Schrödinger semigroups,Simon's problem

Let H = ?Δ + V, where V is a multiplication operator by a real-valued function V(x) on Rⁿ which is uniformly Hölder continuous and $\text{(1 + ¦x¦}{}^2\text{)}{}^{\mathrm{<mtext>?</mtext><mtext>2</mtext>}}\text{V(x) ∈ L}{}_{\mathrm{\infty }}\text{(R}{}^{\mathrm{n}}\text{)}$ for some ? > 4. The relationship between existence of positive solutions, with growth conditions, of Hg = 0 and asymptotic behaviors as t → ∞ of e^?th is established. Using it B. Simon's problem for H on R² is solved.     

A constructive approach to the solution of a class of boundary value problems of mixed type

In this article we discuss the solution of boundary value problems which are described by the linear integrodifferential equation $\text{?xu}\text{?t}\text{(t, x) + u(t, x) ?}\text{1}\text{π}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{∝}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{exp}\text{(?y}{}^2\text{) u(t, y) dy = 0}$, where t∈J?$\text{R}$, x∈$\text{R}$. We interpret the equation in functional form as an ordinary differential equation for the mapping u:J→L₂(R,μ), where L₂(R,μ) is a weighted L₂-space. Emphasis is on the constructive aspects of the solution and on finding representations of the relevant isomorphisms.     

Perturbation theory for eigenvalues and resonances of Schrödinger hamiltonians

Suppose that e^2_?|x|V ∈ ReL^P(R³) for some p > 2 and for g ∈ R, H(g) = ? Δ + gV, H(g) = ?Δ + gV. The main result, Theorem 3, uses Puiseaux expansions of the eigenvalues and resonances of H(g) to study the behavior of eigenvalues λ(g) as they are absorbed by the continuous spectrum, that is $\text{λ(g) ↗6 0}\text{as}\text{g ↘5 g}{}_0\text{> 0}$. We find a series expansion in powers of $\text{(g ? g}{}_0\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{, λ(g) = ∑}{}_{\mathrm{n\; =\; 2}}{}^{\mathrm{\infty }}\text{a}{}_{\mathrm{n}}\text{(g ? g}{}_0\text{)}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$ whose values for g < g₀ correspond to resonances near the origin. These resonances can be viewed as the traces left by the just absorbed eigenvalues.     

Nonlinear “double porosity” type modelUn modèle non linéaire de type double porosité

We consider a variational problem in a bounded domain $\text{Ω=}\text{F}{}^{\mathrm{(\epsilon )}}\text{∪}\text{M}{}^{\mathrm{(\epsilon )}}$ with a microstructure $\text{F}{}^{\mathrm{(\epsilon )}}$ which is not in general periodic; a^ε=a^ε(x) is of order 1 in $\text{F}{}^{\mathrm{(\epsilon )}}$ and as ε→0. A homogenized model is constructed. To cite this article: L. Pankratov, A. Piatnitski, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 435–440.     

Lifting of BV functions with values in S1

We show that for every $\text{u∈}\text{BV}\text{(Ω;S}{}^1\text{)}$, there exists a bounded variation function $\text{?∈}\text{BV}\text{(Ω;}\text{R}\text{)}$ such that u=e^i? a.e. on $\text{Ω}$ and |?|_BV?2|u|_BV. The constant 2 is optimal in dimension n>1. To cite this article: J. Dávila, R. Ignat, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Isolated singularities of some semilinear elliptic equations

Let Ω be an open subset of $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">N</mi>}}$, N ? 3, containing 0. We consider the solutions of ?Δu(x) + g(u(x)) = f(x) in Ω-{0}, where g is nondecreasing and f is bounded and we study the possible singularities at 0: when u(x) = o(|x|^{1 ? N}) we prove that u is isotropic near 0 and show that either it is a C¹ function in Ω (removable singularity) or |x|^{N ? 2}u(x) → c, c ≠ 0 (weak singularity) or |x|^{N ? 2} |u(x) |→ + ∞ (strong singularity). We also characterize the g's for which solutions with a weak singularity exist and improve a previous removability result of H. Brézis and L. Véron (Arch. Rational Mech. Anal.23 (1979), 153–166).     

The asymptotic behavior of solutions of a system of reaction-diffusion equations which models the Belousov-Zhabotinskii chemical reaction

We investigate the boundary value problem $\text{?u}\text{?t}\text{=}\text{?}{}^2\text{u}\text{?x}{}^2\text{+ u(1 ? u ? rv)}$, $\text{?v}\text{?t}\text{=}\text{?}{}^2\text{v}\text{?x}{}^2\text{? buv}$, u(?∞, t) = v(∞, t) = 0, u(∞, t) = 1, and v(?∞, t) = γ ?t > 0 where r > 0, b > 0, γ > 0 and x?R. This system has been proposed by Murray as a model for the propagation of wave fronts of chemical activity in the Belousov-Zhabotinskii chemical reaction. Here u and v are proportional to the concentrations of bromous acid and bromide ion, respectively. We determine the global stability of the constant solution (u, v) ≡ (1,0). Furthermore we introduce a moving coordinate and for each fixed x?R we investigate the asymptotic behavior of u(x + ct, t) and v(x + ct, t) as t → ∞ for both large and small values of the wave speed c ? 0.     

Sur l'équation divu=fOn the equation divu=f

The main result is the following. Let $\text{Ω}$ be a bounded Lipschitz domain in $\text{R}{}^{\mathrm{d}}$, d?2. Then for every $\text{f∈}\text{L}{}^{\mathrm{d}}\text{(Ω)}$ with ∫f=0, there exists a solution $\text{u∈}\text{C}{}^0\text{(}\text{Ω}\text{)∩}\text{W}{}^{\mathrm{1,d}}\text{(Ω)}$ of the equation divu=f in $\text{Ω}$, satisfying in addition u=0 on $\text{?Ω}$ and the estimate

where C depends only on $\text{Ω}$. However one cannot choose u depending linearly on f. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 973–976.     

The sum of aΩ(n) over integers n ≤ x with all prime factors between a and y

For a > 0 let $\text{ψ}{}_{\mathrm{a}}\text{(x, y) =}\text{Σ}\text{a}{}^{\mathrm{\Omega (n)}}$, the sum taken over all n, 1 ≤ n ≤ x such that if p is prime and p|n then a < p ≤ y. It is shown for u < about ($\text{log log}\text{x}\text{log log log}\text{x}$) that $\text{ψ}{}_{\mathrm{a}}\text{(x, x}{}^{\mathrm{<mtext>1</mtext><mtext>u</mtext>}}\text{) ? x(}\text{log}\text{x)}{}^{\mathrm{a?1}}\text{p}{}_{\mathrm{a}}\text{(u)}$, where p_a(u) solves a delay differential equation much like that for the Dickman function p(u), and the asymptotic behavior of p_a(u) is worked out.     

Bifurcation and asymptotics for the Lane–Emden–Fowler equation

We are concerned with the Lane–Emden–Fowler equation ?Δu=λf(u)+a(x)g(u) in $\text{Ω}$, subject to the Dirichlet boundary condition u=0 on $\text{?Ω,}$ where $\text{Ω?}\text{R}{}^{\mathrm{N}}$ is a smooth bounded domain, λ is a positive parameter, $\text{a}\text{:}\text{Ω}\text{→[0,∞)}$ is a Hölder function, and f is a positive nondecreasing continuous function such that f(s)/s is nonincreasing in (0,∞). The singular character of the problem is given by the nonlinearity g which is assumed to be unbounded around the origin. In this Note we discuss the existence and the uniqueness of a positive solution of this problem and we also describe the precise decay rate of this solution near the boundary. The proofs rely essentially on the maximum principle and on elliptic estimates. To cite this article: M. Ghergu, V.D. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Comportement asymptotique semiclassique de l'amplitude de diffusion pour des perturbations captivesSemi-classical behavior of the scattering amplitude for trapping perturbations

We study the semi-classical behavior when h→0, of the scattering amplitude f(θ,ω,λ,h) associated to the Schrödinger operator $\text{P(h)=?}\text{1}\text{2}\text{h}{}^2\text{Δ+V(x)}$ for short range perturbations V(x). We show that if we modify the potential V(x) in a domain contained in {x:V(x)>λ}, the scattering amplitude f(θ,ω,λ,h) is changed by a term of order O(h^∞). Moreover, under an additional escape assumption, we get an asymptotics of the scattering amplitude. To cite this article: L. Michel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 655–660.     

La valeur optimale des programmes entiersThe optimal value of integer programs

We present a formula for the optimal value f_c(y) of the integer program $\text{max}\text{{c′x∣x∈}\text{Ω}\text{(y)∩}\text{N}{}^{\mathrm{n}}\text{}}$ where $\text{Ω}\text{(y)}$ is the convex polyhedron $\text{{x∈}\text{R}{}^{\mathrm{n}}\text{∣Ax=y,}\text{x?0}}$. It is a consequence of Brion and Vergne's formula which evaluates the sum . As in linear programming, f_c(y) can be obtained by inspection of the reduced-costs at the vertices of the polyhedron. We also provide an explicit result that relates f_c(ty) and the optimal value of the associated continous linear program, for large values of $\text{t∈}\text{N}$. To cite this article: J.B. Lasserre, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 863–866.     

Asymptotic behavior for an almost periodic,strongly dissipative wave equation

This paper deals with asymptotic behavior for (weak) solutions of the equation $\text{u}{}_{\mathrm{tt}}\text{? Δu + β(u}{}_{\mathrm{t}}\text{) ? ?(t, x)}$, on $\text{R}$⁺ × Ω; u(t, x) = 0, on $\text{R}$⁺ × ?Ω. If $\text{?∈L∞(R}{}^{\mathrm{+}}\text{,L}{}^2\text{(Ω))}$ and β is coercive, we prove that the solutions are bounded in the energy space, under weaker assumptions than those used by G. Prouse in a previous work. If in addition $\text{?}{}_{\mathrm{t}}\text{∈S}{}^2\text{(R}{}^{\mathrm{+}}\text{,L}{}^2\text{(Ω))}$ and ? is srongly almost-periodic, we prove for strongly monotone β that all solutions are asymptotically almost-periodic in the energy space. The assumptions made on β are much less restrictive than those made by G. Prouse: mainly, we allow β to be multivalued, and in the one-dimensional case β need not be defined everywhere.     

Asymptotics for the blow-up boundary solution of the logistic equation with absorption

Let $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$. Assume that f?0 is a C¹-function on [0,∞) such that f(u)/u is increasing on (0,+∞). Let a be a real number and let b?0, b?0 be a continuous function such that b≡0 on $\text{?Ω}$. The purpose of this Note is to establish the asymptotic behaviour of the unique positive solution of the logistic problem Δu+au=b(x)f(u) in $\text{Ω}$, subject to the singular boundary condition u(x)→+∞ as $\text{dist}\text{(x,?Ω)→0}$. Our analysis is based on the Karamata regular variation theory. To cite this article: F.-C. Cîrstea, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 336 (2003).